Seminars and Colloquia by Series

Polyhedral and tropical geometry in nonlinear algebra

Series
Dissertation Defense
Time
Wednesday, June 30, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Cvetelina HillGeorgia Tech

Please Note: BlueJeans link: https://bluejeans.com/298474885/8484

This dissertation consists of various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. We give a broad introduction and discuss connections to applied algebraic geometry, polyhedral, and tropical geometry. The individual topics discussed are as follows:

  • Interaction between tropical and classical convexity, with a focus on the tropical convex hull of convex sets and polyhedral complexes. Amongst other results, we characterize tropically convex sets in any dimension, and give a combinatorial description for the dimension of the tropical convex hull of an ordinary affine space. 
  • The steady-state degree and mixed volume of a chemical reaction network. We present three case studies of infinite families of networks. For each family, we give a formula for the steady-state degree and mixed volume of the corresponding polynomial system. 
  • Methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. 


Thesis may be viewed here.

BlueJeans link

Algorithmic Approaches to Problems in Probabilistic Combinatorics

Series
Dissertation Defense
Time
Thursday, June 10, 2021 - 10:00 for
Location
ONLINE
Speaker
He GuoGeorgia Institute of Technology

The probabilistic method is one of the most powerful tools in combinatorics: it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method:
1. We study the structural properties of the triangle-free graphs generated by a semirandom variant of triangle-free process and obtain a packing extension of Kim’s famous R(3, t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse.
2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n).
3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory.
4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.

The Bluejeans link of the defense is https://gatech.bluejeans.com/233874892

Homomorphisms and colouring for graphs and binary matroids

Series
Graph Theory Seminar
Time
Tuesday, June 8, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87593953555?pwd=UWl4eTVsanpEUHJDWFo3SWpNNWtxdz09
Speaker
Jim GeelenUniversity of Waterloo

Please Note: Description:This talk is part of the Round the World Relay in Combinatorics

The talk starts with Rödl's Theorem that graphs with huge chromatic number contain triangle-free subgraphs with large chromatic number. We will look at various related results and conjectures, with a notable matroid bias; the new results are joint work with Peter Nelson and Raphael Steiner.

The Density of Costas Arrays Decays Exponentially

Series
Combinatorics Seminar
Time
Friday, May 28, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/6673
Speaker
Christopher SwansonAshland University

Costas arrays are useful in radar and sonar engineering and many other settings in which optimal 2-D autocorrelation is needed: they are permutation matrices in which the vectors joining different pairs of ones are all distinct.
In this talk we discuss some of these applications, and prove that the density of Costas arrays among permutation matrices decays exponentially, solving a core problem in the theory of Costas arrays. 
The proof is probabilistic, and combines ideas from random graph theory with tools from probabilistic combinatorics.

Based on joint work in progress with Bill Correll, Jr. and Lutz Warnke.

Persistence of Invariant Objects under Delay Perturbations

Series
Dissertation Defense
Time
Thursday, May 6, 2021 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE at https://bluejeans.com/137621769
Speaker
Jiaqi YangGeorgia Tech

 We consider functional differential equations which come from adding delay-related perturbations to ODEs or evolutionary PDEs, which is a singular perturbation problem. We prove that for small enough perturbations, some invariant objects (e.g. periodic orbits, slow stable manifolds) of the unperturbed equations persist and depend on the parameters with high regularity. The results apply to state-dependent delay equations and equations which arise in electrodynamics. We formulate results in a posteriori format. The proof is constructive and leads to algorithms. 

This is based on joint works with Joan Gimeno and Rafael de la Llave.

Link: https://bluejeans.com/137621769 

A proof of the Erdős–Faber–Lovász conjecture

Series
School of Mathematics Colloquium
Time
Thursday, May 6, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Tom KellyUniversity of Birmingham

The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$.  In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$.  In this talk, I will present the history of this conjecture and sketch our proof in a special case.

Constructing non-bipartite $k$-common graphs

Series
Graph Theory Seminar
Time
Tuesday, May 4, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Fan WeiPrinceton University

A graph $H$ is $k$-common if the number of monochromatic copies of $H$ in a $k$-edge-coloring of $K_n$ is asymptotically minimized by a random coloring. For every $k$, we construct a connected non-bipartite $k$-common graph. This resolves a problem raised by Jagger, Stovicek and Thomason. We also show that a graph $H$ is $k$-common for every $k$ if and only if $H$ is Sidorenko and that $H$ is locally $k$-common for every $k$ if and only if H is locally Sidorenko.

Normal surface theory and colored Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, May 3, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Christine Ruey Shan LeeUniversity of South Alabama

The colored Jones polynomial is a generalization of the Jones polynomial from the finite-dimensional representations of Uq(sl2). One motivating question in quantum topology is to understand how the polynomial relates to other knot invariants. An interesting example is the strong slope conjecture, which relates the asymptotics of the degree of the polynomial to the slopes of essential surfaces of a knot. Motivated by the recent progress on the conjecture, we discuss a connection from the colored Jones polynomial of a knot to the normal surface theory of its complement. We give a map relating generators of a state-sum expansion of the polynomial to normal subsets of a triangulation of the knot complement. Besides direct applications to the slope conjecture, we will also discuss applications to colored Khovanov homology.

Global Constraints within the Developmental Program of the Drosophila Wing

Series
Mathematical Biology Seminar
Time
Friday, April 30, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Madhav ManiNorthwestern University

Organismal development is a complex process, involving a vast number of molecular constituents interacting on multiple spatio-temporal scales in the formation of intricate body structures. Despite this complexity, development is remarkably reproducible and displays tolerance to both genetic and environmental perturbations. This robustness implies the existence of hidden simplicities in developmental programs. Here, using the Drosophila wing as a model system, we develop a new quantitative strategy that enables a robust description of biologically salient phenotypic variation. Analyzing natural phenotypic variation across a highly outbred population, and variation generated by weak perturbations in genetic and environmental conditions, we observe a highly constrained set of wing phenotypes. Remarkably, the phenotypic variants can be described by a single integrated mode that corresponds to a non-intuitive combination of structural variations across the wing. This work demonstrates the presence of constraints that funnel environmental inputs and genetic variation into phenotypes stretched along a single axis in morphological space. Our results provide quantitative insights into the nature of robustness in complex forms while yet accommodating the potential for evolutionary variations. Methodologically, we introduce a general strategy for finding such invariances in other developmental contexts. -- https://www.biorxiv.org/content/10.1101/2020.10.13.333740v3

Meeting Link: https://gatech.bluejeans.com/348270750

Steady waves in flows over periodic bottoms

Series
CDSNS Colloquium
Time
Friday, April 30, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Carlos Garcia AzpeitiaUNAM

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

In this talk we present the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution for a flat bottom, with the exception of a sequence of velocities $c_{k}$.  Furthermore, we prove that at least two steady solutions for a near-flat bottom persist close to a non-degenerate $S^1$-orbit of steady waves for a flat bottom. As a consequence, we obtain the persistence of at least two steady waves close to a non-degenerate $S^1$-orbit of Stokes waves bifurcating from the velocities $c_{k}$ for a flat bottom. This is a joint work with W. Craig.

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